What is the difference between GDP and GNP? For what types of countries is the difference likely to be important.

Gross domestic product is an attempt to measure production in a country. Gross national product is an attempt to measure the income accruing to residents of a country. Gross domestic product for the US would include, say, income earned from work undertaken in the US by a French consulting engineer, but this would appear as French, not US, gross national product.
For many countries GDP and GNP are similar. Significant differences arise when there is a lot of income earned from assets owned abroad, and this is most often true of small countries.